Q1. Which of the following are sets ? Justify your answer.
(i) The collection of all the months of a year beginning with the letter J.
(ii) The collection of ten most talented writers of India.
(iii) A team of eleven best-cricket batsmen of the world.
(iv) The collection of all boys in your class.
(v) The collection of all natural numbers less than 100.
(vi) A collection of novels written by the writer Munshi Prem Chand.
(vii) The collection of all even integers.
(viii) The collection of questions in this Chapter.
(ix) A collection of most dangerous animals of the world.

Solution

Solution: A collection is called a set if its members are well-defined, meaning that there is no ambiguity about whether an object belongs to the collection or not.

(i) The collection of all the months of a year beginning with the letter J is a set because the months satisfying this condition are clearly identified. In fact, the elements can be written as \[ \begin{aligned} J = \{\text{January}, \text{June}, \text{July}\} \end{aligned} \] Hence, the collection is well-defined.

(ii) The collection of ten most talented writers of India is not a set because the term “most talented” is subjective and may differ from person to person. Therefore, the collection is not well-defined.

(iii) A team of eleven best-cricket batsmen of the world is not a set since the word “best” has no fixed or universally accepted criterion. Different opinions may lead to different selections, making the collection ill-defined.

(iv) The collection of all boys in your class is a set because the members of the class are fixed and it is always possible to decide whether a particular student is a boy in that class or not. Hence, the collection is well-defined.

(v) The collection of all natural numbers less than 100 is a set, as the elements are clearly specified and can be written as \[ \begin{aligned} N = \{1, 2, 3, \ldots, 99\} \end{aligned} \] Thus, there is no ambiguity in membership.

(vi) A collection of novels written by the writer Munshi Prem Chand is a set because the novels authored by him are definite and known, making the collection well-defined.

(vii) The collection of all even integers is a set since the condition for membership is precise. Symbolically, it can be expressed as \[ \begin{aligned} E = \{2n \mid n \in \mathbb{Z}\} \end{aligned} \]

(viii) The collection of questions in this Chapter is a set because the chapter contains a fixed and definite number of questions, and each question can be clearly identified.

(ix) A collection of most dangerous animals of the world is not a set because the word “most dangerous” is vague and depends on individual judgment, making the collection ill-defined.

Q2. Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces:
(i) 5 . . .A
(ii) 8 . . . A
(iii) 0. . .A
(iv) 4. . . A
(v) 2. . .A
(vi) 10. . .A

Solution

Solution: The given set is

\( \begin{aligned} A = \{1, 2, 3, 4, 5, 6\}. \end{aligned} \)

We examine each case to determine whether the given element belongs to the set \(A\) or not.

\( \begin{aligned} \text{(i)}\;& 5 \in A \quad \text{since } 5 \text{ is an element of } A.\\ \text{(ii)}\;& 8 \notin A \quad \text{because } 8 \text{ is not present in the set } A.\\ \text{(iii)}\;& 0 \notin A \quad \text{as } 0 \text{ does not belong to } A.\\ \text{(iv)}\;& 4 \in A \quad \text{since } 4 \text{ is clearly an element of } A.\\ \text{(v)}\;& 2 \in A \quad \text{because } 2 \text{ is included in the set } A.\\ \text{(vi)}\;& 10 \notin A \quad \text{as } 10 \text{ is not an element of } A. \end{aligned} \)

Q3. Write the following sets in roster form:
(i) A = {x : x is an integer and –3 ≤ x < 7}
(ii) B = {x : x is a natural number less than 6}
(iii) C = {x : x is a two-digit natural number such that the sum of its digits is 8}
(iv) D = {x : x is a prime number which is divisor of 60}
(v) E = The set of all letters in the word TRIGONOMETRY
(vi) F = The set of all letters in the word BETTER

Solution

Solution: Each set is expressed in roster form by listing all distinct elements that satisfy the given condition.

\( \begin{aligned} A &= \{-3, -2, -1, 0, 1, 2, 3, 4, 5, 6\} \end{aligned}\)
since all integers from -3 to 6 are included.

\( \begin{aligned}B &= \{1, 2, 3, 4, 5\} \end{aligned} \)
As the natural numbers less than 6 are 1 to 5

\( \begin{aligned}C &= \{17, 26, 35, 44, 53, 62, 71, 80\} \end{aligned} \)
because each two-digit number has a digit sum of 8

\(\begin{aligned}D &= \{2, 3, 5\} \end{aligned} \)
since these are the prime divisors of 60

\(\begin{aligned}E &= \{T, R, I, G, O, N, M, E, Y\} \end{aligned} \)
As repeated letters are written only once in a set

\(\begin{aligned}F &= \{B, E, T, R\} \end{aligned} \)
because the word BETTER contains these distinct letters.

Q4. Write the following sets in the set-builder form :
(i) {3, 6, 9, 12}
(ii) {2,4,8,16,32}
(iii) {5, 25, 125, 625}
(iv) {2, 4, 6, . . .}
(v) {1,4,9, . . .,100}

Solution

Solution: Each given set is rewritten in set-builder form by identifying the common rule followed by its elements.

\[ \begin{aligned} A &= \{3, 6, 9, 12\} \\ A &= \{\, x : x = 3n,\; n \in \mathbb{N},\; 1 \le n \le 4 \,\}. \end{aligned} \]

\[ \begin{aligned} B &= \{2, 4, 8, 16, 32\} \\ B &= \{\, x : x = 2^{n},\; n \in \mathbb{N},\; 1 \le n \le 5 \,\}. \end{aligned} \]

\[ \begin{aligned} C &= \{5, 25, 125, 625\} \\ C &= \{\, x : x = 5^{n},\; n \in \mathbb{N},\; 1 \le n \le 4 \,\}. \end{aligned} \]

\[ \begin{aligned} D &= \{2, 4, 6, \ldots\} \\ D &= \{\, x : x = 2n,\; n \in \mathbb{N},\; n \ge 1 \,\}. \end{aligned} \]

\[ \begin{aligned} E &= \{1, 4, 9, \ldots, 100\} \\ E &= \{\, x : x = n^{2},\; n \in \mathbb{N},\; 1 \le n \le 10 \,\}. \end{aligned} \]

Q5. List all the elements of the following sets :
(i) \(\mathrm{A = \{x : x\text{ is an odd natural number}\}}\)
(ii) \(\mathrm{B = \{x : x\text{ is an integer}, -\frac{1}{2} \lt x \lt \frac{9}{2}\}}\) (iii) \(\mathrm{C = \{x : x\text{ is an integer}, x^2 ≤ 4\}}\) (iv) \(\mathrm{D = \{x : x\text{ is a letter in the word “LOYAL”}\}}\) (v) \(\mathrm{E = {x : x\text{ is a month of a year not having 31 days}}}\) (v0) \(\mathrm{\{x : x\text{ is a consonant in the English alphabet which precedes k }\}}\)

Solution

Solution: Each set is written by listing all elements that satisfy the given condition, ensuring that no repetition occurs and every element is well-defined.

\( \begin{aligned} A &= \{1, 3, 5, \ldots\} \end{aligned} \)
since the set consists of all odd natural numbers

\( \begin{aligned}B &= \{0, 1, 2, 3, 4\} \end{aligned} \)
as these are the integers lying between \(-\tfrac{1}{2}\) and \(\tfrac{9}{2}\)

\( \begin{aligned}C &= \{-2, -1, 0, 1, 2\} \end{aligned} \)
because \(x^2 \le 4\) for these integer values only

\( \begin{aligned}D &= \{L, O, Y, A\} \end{aligned} \)
since these are the distinct letters in the word “LOYAL”

\( \begin{aligned}E &= \{\text{February}, \text{April}, \text{June}, \text{September}, \text{November}\} \end{aligned} \)
as these months do not have 31 days.

\( \begin{aligned}F &= \{B, C, D, F, G, H, J\} \end{aligned} \)
because these consonants precede the letter k in the English alphabet.

Q6. Match each of the set on the left in the roster form with the same set on the right described in set-builder form: \( \begin{array}{l} (i)\; \{1, 2, 3, 6\}& (a)\; \{x : x \text{ is a prime number and a divisor of }6\}\\ (ii)\; \{2, 3\} & (b)\; \{x : x \text{ is an odd natural number less than} 10\}\\ (iii)\; \{M,A,T,H,E,I,C,S\}&(c)\; \{x : x \text{ is natural number and divisor of }6\}\\ (iv)\; \{1, 3, 5, 7, 9\}&(d)\; \{x : x \text{ is a letter of the word MATHEMATICS\}} \end{array} \)

Solution

Solution: To match each set written in roster form with its corresponding set-builder description, we carefully interpret the meaning of each set-builder form and compare it with the listed elements.

\( \begin{aligned} \text{(i)}\;& \{1, 2, 3, 6\} \\ &\text{These are all natural numbers which divide } 6. \\ &\Rightarrow \text{It matches with } (c). \end{aligned} \)

\( \begin{aligned} \text{(ii)}\;& \{2, 3\} \\ &\text{These are the prime numbers which divide } 6. \\ &\Rightarrow \text{It matches with } (a). \end{aligned} \)

\( \begin{aligned} \text{(iii)}\;& \{M, A, T, H, E, I, C, S\} \\ &\text{These are the distinct letters of the word MATHEMATICS.} \\ &\Rightarrow \text{It matches with } (d). \end{aligned} \)

\( \begin{aligned} \text{(iv)}\;& \{1, 3, 5, 7, 9\} \\ &\text{These are odd natural numbers less than } 10. \\ &\Rightarrow \text{It matches with } (b). \end{aligned} \)

Hence, the correct matching is: \[ \begin{aligned} (i) \rightarrow (c), \\ (ii) \rightarrow (a), \\ (iii) \rightarrow (d), \\ (iv) \rightarrow (b). \end{aligned} \]